We prove the “star” conjecture restricted to homoclinic classes. To be
precise, for
C^1-generic diffeomorphisms, if the periodic orbits contained in a
homoclinic class H(p)
have all their Lyapunov exponents bounded away from 0, then H(p) must
be (uniformly)
hyperbolic. This gives a way to characterize lack of hyperbolicity of homoclinic
classes through the existence of “weak” periodic orbits. The main
difficulty to be “restricted”
is that the homoclinic class H(p) is not known isolated in advance. Hence
the “weak” periodic orbits created by perturbations near the
homoclinic class have to
be guaranteed strictly inside the homoclinic class. We construct in
the proof several
perturbations which are not simple applications of the connecting lemmas.

We give a short introduction to set-valued numerics - the
foundation of rigorous computations. These techniques are applied to the
classical problem in Newtonian n-body of determining the number of
relative equlibria a set of planets can display. Some partial results
for the restricted 3-body problem are discussed, as well as future
goals. This is joint work with Piotr Zgliczynski.

(講演2) In search for the H&eacutenon attractor

Abstract:

By performing a systematic study of the Hénon map, we find
low-period sinks for parameter values extremely close to the classical
ones. This raises the question whether or not the well-known Hénon
attractor is a strange attractor, or simply a stable periodic orbit.
Using results from our study, we conclude that even if the latter were
true, it would be practically impossible to establish this by computing
trajectories of the map. This is joint work with Zbigniew Galias.

The escape rate of asteroids, chemical reaction rates, and fluid mixing rates are
all examples of chaotic transport rates. One can typically launch
a Monte Carlo simulation over millions of trajectories and compute
such rates, however we seek to extract this information from a smaller
number of trajectories.
The homotopic lobe dynamics technique uses the topological forcing of
stable and unstable manifolds of saddle points to compute a system's
symbolic dynamics, which can be used to compute periodic orbits and
transport rates. I will demonstrate this technique by computing mixing
rates in a fluid system and the ionization rate for a classical atomic
system.

5月27日（金）

場所, 時間：東京大学駒場キャンパス 数理科学研究科棟 122号室 15:00-18:00

松元 重則 氏（日本大学）

Dynamics of the horocycle and geodesic flows of compact foliations by hyperbolic surfaces

In a recent paper by Kiriki and Soma a solution to van Strien problem of
the existence of non-trivial wandering domains for two-dimensional
maps was proposed
for the case of finite smoothness. We discuss a possibility to extend
their results
to the real-analytic case and the case of three-dimensional polynomial
diffeomorphisms.